Peering Past the Frontier
As we move on to other systems of Z-equations, we ﬁnd that
particular examples tend to be less and less of interest. Do you
really care exactly how many solutions to the system
− 42xy + z
− 34xyz + z
x + y
there are with x, y, and z all integers? Unless this problem has
some burning practical signiﬁcance, it is not very interesting
merely on its own. We just made it up at random. On the other
hand, it does represent the kind of problem that interests us in
general, namely, ﬁnding the solutions to a system of Z-equations.
Any adequate theory we develop should eventually be able to tell
us about this random system and all other systems too.
Thus, our focus tends to shift to whole classes of problems, and
therefore to the structural or qualitative statements we can make
or prove about them. For example, what computable properties of
a system of Z-equations can guarantee that there are only ﬁnitely
many solutions with the variables taking on only rational values?
Such questions, which are at the frontier of research, lead back to
more structural questions about varieties and the absolute Galois
group of Q.
For another example, consider the statement: The absolute
Galois group of Q is big. Now, make it into a precise assertion and
prove it. That might include proving that for any positive integer n
and any prime p, there is a Galois representation whose image
all of GL(n, F
). (It is not known if this is true.) But there will be
many other ways of envisioning the “bigness” of the absolute Galois
group of Q, according to various other aspects of its structure.
In this way, after we leave behind particular Z-equations or sys-
tems that have provoked our study in the ﬁrst place, the structures
The image of a group representation φ is the set of all elements in the target that are
actually values of φ.